# Maple code
# Output: lifting square system of equations for the Schubert problem
#
# (489123567)^4 * (689123457)^6 which is written compactly as
#
# [][][]^4 []^6 = 437 in Gr(3,9)
#
# Nickolas Hein
# 27 March 2015
#
# This is a proof of concept for computing a Schubert problem via lifting in bertini.
# Therefore, precomputations involving linear algebra are omitted.
# In particular, we randomly choose dual flags rather than primal flags.
#
with(LinearAlgebra):
with(combinat):
k:=3:
n:=9:
# Given an instance
#
# [][][]F1 n ... n [][][]F4 n []F5 n ... n []F10
#
# of the Schubert problem, we choose a basis of the ambient space so that F1 is the coordinate flag and F2 is the opposite coordinate flag.
# So [][][]F1 n [][][]F2 is locally the row space of the 9x3 matrix C with columns c[i]:
c[1]:=Matrix(n,1,[[ x1 ],[ x2 ],[ x3 ],[ 1 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ]]):
c[2]:=Matrix(n,1,[[ 0 ],[ x4 ],[ x5 ],[ x6 ],[ x7 ],[ x8 ],[ x9 ],[ 1 ],[ 0 ]]):
c[3]:=Matrix(n,1,[[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ 0 ],[ x10 ],[ x11 ],[ x12 ],[ 1 ]]):
# We place these entries in a square matrix M so that M=[C|0].

# The additional columns will be used later to calculate determinantal hypersurface conditions.
M:=Matrix(n,n):
for i from 1 to n do
 for j from 1 to k do
  M[i,j]:=c[j][i,1]:
 od:
od:

# The lifting formulation introduces some new variables.
# For the F3 condition
b[1]:=b12:
b[2]:=b13:
# For the F4 condition
b[3]:=c12:
b[4]:=c13:

fprintf(file,"variable_group x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12;\n"):
fprintf(file,"variable_group b12,b13,c12,c13;\n\n"):

# We separate the functions into bilinear equations Bi and determinantal equations Di.
fprintf(file,"function B1,B2,B3,B4,B5,B6,B7,B8,B9,B10;\n"):
fprintf(file,"function D1,D2,D3,D4,D5,D6;\n\n"):

# Go through the following loop once for each [][][] condition
for i from 0 to 1 do
 # Since we use lifting to obtain bilinear equations for non-hypersurface Schubert varieties with defining flags F, we randomly select flags G dual to F (randomly choosing G rather than F avoids a (simple) precomputation).
 # The flag G has ordered basis of rows g1,...,gn.
 for j from 1 to 5 do
  g[j]:=RandomMatrix(1,n):
  # Here we also multiply by 1/100 using notation familiar to bertini. This helps condition the system numerically.
  fprintf(file,"B%a = 1e-2*(%a);\n\n",i*5+j,simplify(Multiply(g[j],c[1]+b[2*i+1]*c[2]+b[2*i+2]*c[3])[1,1])):
 od:
od:
# We randomly choose flags F for the determinantal conditions of []F by choosing random nxn matrices N (say columns N_n,N_(n-1),...,N_1 form a basis for F).
# Here are the determinantal conditions:
for ii from 1 to 6 do # One eqn for each [] condition
 N:=RandomMatrix(n,n):
 for i from 1 to n do
  for j from k+1 to n do
   M[i,j]:=N[i,j]:
  od:
 od:
 fprintf(file,"D%a = 1e-11*(%a);\n\n",ii,simplify(Determinant(M))):
 # Again, we scaled to condition the system.
od:
fclose(file):

CONFIG

USEREGENERATION: 1;
TRACKTOLBEFOREEG: 1e-7;
TRACKTOLDURINGEG: 1e-7;
FINALTOL: 1e-11;
SLICETOLBEFOREEG: 1e-7;
SLICETOLDURINGEG: 1e-7;
SLICEFINALTOL: 1e-11;
SECURITYMAXNORM: 1e8;
MAXNORM: 1e8;
PRINTPATHMODULUS: 1000;
SAMPLEFACTOR: 0.1;
NUMSAMPLEPOINTS: 3;
ODEPREDICTOR: 2;
RANDOMSEED: 1000;

END;
INPUT

variable_group x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12;
variable_group b12,b13,c12,c13;

function B1,B2,B3,B4,B5,B6,B7,B8,B9,B10;
function D1,D2,D3,D4,D5,D6;

B1 = 1e-2*(-21*x1-56*x2-56*b12*x4-8*x3-8*b12*x5-50-50*b12*x6+30*b12*x7+62*b12*x8+62*b13*x10-79*b12*x9-79*b13*x11-71*b12-71*b13*x12+28*b13);

B2 = 1e-2*(66*x1-36*x2-36*b12*x4-41*x3-41*b12*x5+20+20*b12*x6-7*b12*x7+16*b12*x8+16*b13*x10-34*b12*x9-34*b13*x11-62*b12-62*b13*x12-90*b13);

B3 = 1e-2*(97*x1-82*x2-82*b12*x4-66*x3-66*b12*x5+55+55*b12*x6+68*b12*x7+26*b12*x8+26*b13*x10+13*b12*x9+13*b13*x11-65*b12-65*b13*x12+5*b13);

B4 = 1e-2*(-42*x1-82*x2-82*b12*x4+59*x3+59*b12*x5-69-69*b12*x6+23*b12*x7+25*b12*x8+25*b13*x10+5*b12*x9+5*b13*x11-75*b12-75*b13*x12+38*b13);

B5 = 1e-2*(22*x1-81*x2-81*b12*x4-99*x3-99*b12*x5-2-2*b12*x6-98*b12*x7-41*b12*x8-41*b13*x10+24*b12*x9+24*b13*x11+65*b12+65*b13*x12+b13);

B6 = 1e-2*(-90*x1-66*x2-66*c12*x4-11*x3-11*c12*x5+35+35*c12*x6+61*c12*x7+96*c12*x8+96*c13*x10+53*c12*x9+53*c13*x11+61*c12+61*c13*x12-70*c13);

B7 = 1e-2*(22*x1-34*x2-34*c12*x4+69*x3+69*c12*x5+55+55*c12*x6-12*c12*x7+98*c12*x8+98*c13*x10+56*c12*x9+56*c13*x11-54*c12-54*c13*x12-88*c13);

B8 = 1e-2*(89*x1-70*x2-70*c12*x4-44*x3-44*c12*x5-48-48*c12*x6-52*c12*x7+37*c12*x8+37*c13*x10+58*c12*x9+58*c13*x11-45*c12-45*c13*x12+72*c13);

B9 = 1e-2*(-99*x1+14*x2+14*c12*x4+42*x3+42*c12*x5-68-68*c12*x6-19*c12*x7+51*c12*x8+51*c13*x10-35*c12*x9-35*c13*x11-52*c12-52*c13*x12+71*c13);

B10 = 1e-2*(-30*x1-52*x2-52*c12*x4+9*x3+9*c12*x5-43-43*c12*x6+78*c12*x7+72*c12*x8+72*c13*x10+50*c12*x9+50*c13*x11+32*c12+32*c13*x12-23*c13);

D1 = 1e-13*(-2268558196005*x10*x6*x2-887950513743*x10*x2*x5-21945679458*x2*x12*x5-127241229351*x12*x6*x3+375315441243*x12*x6*x1-602961721617*x10*x6*x3-1850147919432*x11*x6*x2-52860294111*x12*x7*x3+126257046423*x1*x12*x7-208466973822*x11*x2*x5-682584346911*x12*x8*x2+208466973822*x11*x4*x3+21945679458*x12*x4*x3+887950513743*x10*x4*x3-18028539072*x12*x9*x3+79667277525*x1*x12*x5-156131724789*x1*x7*x11-180201193851*x12*x7*x2+2263246576122*x10*x9*x2-192972339819*x12*x8*x3+550726492167*x1*x12*x8-784038536457*x11*x6*x3-462893849673*x1+588815994438*x2+167682630036*x3+1146961530444*x4+264121148181*x5+8116860900*x7+178750045431*x8+794315995572*x9-232327548015*x11-857522584311*x1*x7*x10+29502488856*x10+219169076631*x1*x12*x4-726769455906*x2*x5-104316409362*x12*x9*x2-453434537211*x12*x6*x2-471725056992*x7*x11*x2+1103629952709*x10*x9*x3+1343166293097*x1*x10*x5-2263246576122*x8*x11*x2-1103629952709*x8*x11*x3+50644353801-1146961530444*x6*x2-264121148181*x6*x3+1032488765571*x6*x1+127241229351*x12*x5-47374316340*x12*x7-29502488856*x12*x8+232327548015*x12*x9+453434537211*x12*x4+726769455906*x4*x3+2138676476808*x9*x2+995808717420*x9*x3+104316409362*x11*x2+18028539072*x11*x3+1850147919432*x11*x4+784038536457*x11*x5+1219283750046*x8*x2+394039978923*x8*x3+682584346911*x10*x2+192972339819*x10*x3+2268558196005*x10*x4+602961721617*x10*x5-1282726576497*x8*x11+1282726576497*x10*x9+1031791608876*x7*x2+242743164099*x7*x3-287820134097*x7*x11*x3-860567419200*x1*x8-550726492167*x1*x10-780263635980*x7*x11+155136206823*x7*x10*x3-162479980140*x7*x10+822953891655*x7*x10*x2-918050317209*x1*x7+198640743021*x1*x12*x9+581111882811*x1*x11*x5+1150048736631*x6*x1*x11+1520889402486*x1*x4+1004462942343*x1*x5-871949606904*x1*x9-198640743021*x1*x11+1065503444310*x1*x11*x4+1926150363309*x6*x1*x10+2216931274794*x1*x10*x4+668105651016*x1*x8*x11-668105651016*x1*x10*x9);

D2 = 1e-12*(-35215492620*x10*x6*x2+63914723946*x10*x2*x5-88152548630*x2*x12*x5+37218893545*x12*x6*x3+57068777445*x12*x6*x1-44458425575*x10*x6*x3-113861559280*x11*x6*x2-248247637785*x12*x7*x3+30282109007*x1*x12*x7+193308509752*x11*x2*x5-68747412324*x12*x8*x2-193308509752*x11*x4*x3+88152548630*x12*x4*x3-63914723946*x10*x4*x3+697176322627*x12*x9*x3-81110532316*x1*x12*x5+131559515685*x1*x7*x11+58712628550*x12*x7*x2-52738655208*x10*x9*x2+172844518996*x12*x8*x3-46726183300*x1*x12*x8-152094890800*x11*x6*x3+42367104281*x1+94372459048*x2-464977161147*x3+147445652200*x4+147474711390*x5+205070522900*x7+152816118290*x8+632226590260*x9+517011752230*x11+1427753347*x1*x7*x10+143053212670*x10+8430160654*x1*x12*x4+237961667532*x2*x5-135186580248*x12*x9*x2+58155171620*x12*x6*x2+151501548640*x7*x11*x2-126457101805*x10*x9*x3-103030047036*x1*x10*x5+52738655208*x8*x11*x2+126457101805*x8*x11*x3+346595368910-147445652200*x6*x2-147474711390*x6*x3+174756312970*x6*x1-37218893545*x12*x5+188560663415*x12*x7-143053212670*x12*x8-517011752230*x12*x9-58155171620*x12*x4-237961667532*x4*x3+157978695568*x9*x2-862336930554*x9*x3+135186580248*x11*x2-697176322627*x11*x3+113861559280*x11*x4+152094890800*x11*x5+117154402992*x8*x2-129451538787*x8*x3+68747412324*x10*x2-172844518996*x10*x3+35215492620*x10*x4+44458425575*x10*x5-32990388250*x8*x11+32990388250*x10*x9+37150353340*x7*x2-293804431041*x7*x3-900918655351*x7*x11*x3-70857893199*x1*x8+46726183300*x1*x10+649855583390*x7*x11-178328165773*x7*x10*x3+167778080685*x7*x10+99949007370*x7*x10*x2+47215844843*x1*x7-57715257937*x1*x12*x9-474918803512*x1*x11*x5+364781175680*x6*x1*x11+65606844036*x1*x4-216418407600*x1*x5+94072742942*x1*x9+57715257937*x1*x11+108092453800*x1*x11*x4+97290627845*x6*x1*x10+58257793134*x1*x10*x4-200279012623*x1*x8*x11+200279012623*x1*x10*x9);

D3 = 1e-13*(1465904817648*x10*x6*x2+182993389836*x10*x2*x5-1070020808316*x2*x12*x5+3633535638036*x12*x6*x3-519121644279*x12*x6*x1-1541584274580*x10*x6*x3+2613856403076*x11*x6*x2-3956783744232*x12*x7*x3-1789272299962*x1*x12*x7-2263855445736*x11*x2*x5+895079326834*x12*x8*x2+2263855445736*x11*x4*x3+1070020808316*x12*x4*x3-182993389836*x10*x4*x3-3527721808212*x12*x9*x3-3267866728224+1591616774640*x1*x12*x5+485923188683*x1*x7*x11+874471810494*x12*x7*x2-2013826107257*x10*x9*x2-522224451492*x12*x8*x3-1395874589797*x1*x12*x8+1471513324392*x11*x6*x3+1049230729673*x1-483326342344*x2+2675238585108*x3-1770126480168*x4-415779663564*x5-1333322005386*x7+1770579699906*x8-805111622571*x9-4529580989043*x11-1712941496299*x1*x7*x10-3357000359214*x10+132110983051*x1*x12*x4-1510608704340*x2*x5+702241926833*x12*x9*x2+650596891800*x12*x6*x2+83968862233*x7*x11*x2+1708182145884*x10*x9*x3-1273794634296*x1*x10*x5+2013826107257*x8*x11*x2-1708182145884*x8*x11*x3+1770126480168*x6*x2+415779663564*x6*x3+3685930474191*x6*x1-3633535638036*x12*x5+5375312002074*x12*x7+3357000359214*x12*x8+4529580989043*x12*x9-650596891800*x12*x4+1510608704340*x4*x3-31184631829*x9*x2+679749556188*x9*x3-702241926833*x11*x2+3527721808212*x11*x3-2613856403076*x11*x4-1471513324392*x11*x5+1346291713934*x8*x2-1194769082448*x8*x3-895079326834*x10*x2+522224451492*x10*x3-1465904817648*x10*x4+1541584274580*x10*x5+3281267275023*x8*x11-3281267275023*x10*x9+18959679122*x7*x2+1142297400036*x7*x3+500921847852*x7*x11*x3+1689592484064*x1*x8+1395874589797*x1*x10-523786305381*x7*x11+1990094844756*x7*x10*x3-4282114886052*x7*x10-2400704522092*x7*x10*x2+878209391353*x1*x7-1480112175677*x1*x12*x9-3398580338376*x1*x11*x5+4981160826696*x6*x1*x11+1113502690057*x1*x4-2883989896344*x1*x5+441521880781*x1*x9+1480112175677*x1*x11+1626371331540*x1*x11*x4+4067735093427*x6*x1*x10+1694121020741*x1*x10*x4+1796056081927*x1*x8*x11-1796056081927*x1*x10*x9);

D4 = 1e-13*(-1471221581162*x10*x6*x2+2004530765038*x10*x2*x5+469944702394*x2*x12*x5+225587422784*x12*x6*x3-259892779822*x12*x6*x1+4745572568*x10*x6*x3-757033405742*x11*x6*x2-42297159138*x12*x7*x3+117739212556*x1*x12*x7+1301457877858*x11*x2*x5-2551356822480*x12*x8*x2-1301457877858*x11*x4*x3-469944702394*x12*x4*x3-2004530765038*x10*x4*x3-577755762944*x12*x9*x3+40427375578*x1*x12*x5-1110090965475*x1*x7*x11+562401353788*x12*x7*x2+311358268422*x10*x9*x2-352689124270*x12*x8*x3+358851214240*x1*x12*x8-943766533312*x11*x6*x3-585674115070*x1+554702121090*x2+386266153828*x3+475739561326*x4+866934884336*x5+304697613310*x7-1001484531514*x8-162493206444*x9+812218487360*x11-891912193918*x1*x7*x10-264895660010*x10+770607509200*x1*x12*x4+378538216274*x2*x5-1583493914094*x12*x9*x2+1278938495794*x12*x6*x2-7185410816*x7*x11*x2-1487662609498*x10*x9*x3-773918021129*x1*x10*x5-311358268422*x8*x11*x2+1487662609498*x8*x11*x3-475739561326*x6*x2-866934884336*x6*x3-725535869816*x6*x1-225587422784*x12*x5-385079640506*x12*x7+264895660010*x12*x8-812218487360*x12*x9-1278938495794*x12*x4-378538216274*x4*x3+260687019156*x9*x2+604340237172*x9*x3+1583493914094*x11*x2+577755762944*x11*x3+757033405742*x11*x4+943766533312*x11*x5+784936674388+310954471350*x8*x2+1363513778486*x8*x3+2551356822480*x10*x2+352689124270*x10*x3+1471221581162*x10*x4-4745572568*x10*x5-1091130054710*x8*x11+1091130054710*x10*x9-95103930472*x7*x2-415749754882*x7*x3-688032563498*x7*x11*x3+2300818688755*x1*x8-358851214240*x1*x10+395004935078*x7*x11-528918141586*x7*x10*x3+388487680518*x7*x10-122160784364*x7*x10*x2-700773173577*x1*x7+811172940722*x1*x12*x9-1438314529406*x1*x11*x5-696951554734*x6*x1*x11-867693732145*x1*x4-1003891519936*x1*x5+693934553226*x1*x9-811172940722*x1*x11-2114829428327*x1*x11*x4+576441280811*x6*x1*x10-3558977688440*x1*x10*x4+2761506037541*x1*x8*x11-2761506037541*x1*x10*x9);

D5 = 1e-12*(173290033230*x10*x6*x2-174531233642*x10*x2*x5-188558597488*x2*x12*x5-43864737280*x12*x6*x3-146440079940*x12*x6*x1-173233561242*x10*x6*x3+179298175474*x11*x6*x2-213899949096*x12*x7*x3-198337107056*x1*x12*x7-875840251878*x11*x2*x5+66116651970*x12*x8*x2+875840251878*x11*x4*x3+188558597488*x12*x4*x3+174531233642*x10*x4*x3+267060278932*x12*x9*x3+435687317128*x1*x12*x5+140455713732*x1*x7*x11-17000724436*x12*x7*x2-314919487108*x10*x9*x2-65369449684*x12*x8*x3-225388747458*x1*x12*x8+338106428026*x11*x6*x3-299313133108*x1+162746152360*x2+69072024736*x3+440570160160*x4+25664371408*x5-307587601464*x7-211117540140*x8+377329704492*x9+57929797450*x11-832914706548*x1*x7*x10+720495510*x10+209467870116*x1*x12*x4+492108187608*x2*x5-8440619282*x12*x9*x2+42287857080*x12*x6*x2-109007876662*x7*x11*x2+550829061092*x10*x9*x3+526782960942*x1*x10*x5+314919487108*x8*x11*x2-550829061092*x8*x11*x3-53350657360-440570160160*x6*x2-25664371408*x6*x3+888680525304*x6*x1+43864737280*x12*x5+51926048020*x12*x7-720495510*x12*x8-57929797450*x12*x9-42287857080*x12*x4-492108187608*x4*x3+777972104622*x9*x2-376150922046*x9*x3+8440619282*x11*x2-267060278932*x11*x3-179298175474*x11*x4-338106428026*x11*x5-21914986030*x8*x2+234537526718*x8*x3-66116651970*x10*x2+65369449684*x10*x3-173290033230*x10*x4+173233561242*x10*x5+234333770472*x8*x11-234333770472*x10*x9-534856861004*x7*x2+312412470028*x7*x3-43060824130*x7*x11*x3+158822506956*x1*x8+225388747458*x1*x10-33265811364*x7*x11-430642273766*x7*x10*x3+209634062004*x7*x10+219578251486*x7*x10*x2+1245859476840*x1*x7+316177673066*x1*x12*x9+1519156606298*x1*x11*x5+70760313888*x6*x1*x11-239189202356*x1*x4-1006572036188*x1*x5-1774114364364*x1*x9-316177673066*x1*x11+988909737266*x1*x11*x4-591489615024*x6*x1*x10+68965709094*x1*x10*x4-1168172208552*x1*x8*x11+1168172208552*x1*x10*x9);

D6 = 1e-13*(714333598632*x10*x6*x2-346842021387*x10*x2*x5-210494246778*x2*x12*x5+198755124372*x12*x6*x3+897302934120*x12*x6*x1-314648922948*x10*x6*x3+120408728478*x11*x6*x2+30325074072*x12*x7*x3+124954350828*x1*x12*x7+191462843031*x11*x2*x5-290122762582*x12*x8*x2-191462843031*x11*x4*x3+210494246778*x12*x4*x3+346842021387*x10*x4*x3-17951239380*x12*x9*x3-30421746468*x1*x12*x5-103398278960*x1*x7*x11+61207647902*x12*x7*x2+293018143783*x10*x9*x2-221848114638*x12*x8*x3-891339604356*x1*x12*x8-232745633754*x11*x6*x3+257094506136*x1-139250261260*x2+72404022732*x3-974591042958*x4+89137689150*x5+634525167022*x7+398796188124*x8+354676960652*x9-399567609272*x11-1340803604139*x1*x7*x10-720098377496*x10+805937962584*x1*x12*x4-127551994971*x2*x5+337982006768*x12*x9*x2+943167588684*x12*x6*x2+538028231203*x7*x11*x2-231369339771*x10*x9*x3-236077360863*x1*x10*x5-293018143783*x8*x11*x2+231369339771*x8*x11*x3+974591042958*x6*x2-89137689150*x6*x3+181698541700*x6*x1-198755124372*x12*x5-78084287668*x12*x7+720098377496*x12*x8+399567609272*x12*x9-943167588684*x12*x4+127551994971*x4*x3+331465259546*x9*x2-76735572024*x9*x3-337982006768*x11*x2+17951239380*x11*x3-120408728478*x11*x4+232745633754*x11*x5-405253732839*x8*x2-15128261463*x8*x3+290122762582*x10*x2+221848114638*x10*x3-714333598632*x10*x4+314648922948*x10*x5-210692607124*x8*x11+210692607124*x10*x9+281697779029*x7*x2-108809577864*x7*x3-59116586664*x7*x11*x3-323151418174*x1*x8+891339604356*x1*x10+616858951570*x7*x11-339732026274*x7*x10*x3+1070525583404*x7*x10-408777043285*x7*x10*x2-341434469272*x1*x7-19884545232*x1*x12*x9+12624755376*x1*x11*x5-538819866440*x6*x1*x11+605087191358*x1*x4-79122505800*x1*x5-158408732352*x1*x9+19884545232*x1*x11-449779068092*x1*x11*x4-497908770380*x6*x1*x10+1084808020087*x1*x10*x4+524205125404*x1*x8*x11-524205125404*x1*x10*x9-455907142336);

END;

[linux]:~$ time

real 25m53.369s
user 71m27.557s
sys 0m9.072s

3.6 GHz processors used. Total processing power:

4.296629 Ghz-hours